How long would it take each of them to complete the cleaning alone?

Question

Genny and Ty work together to clean the rooms in a B&B in 2 2/9 hours.

 

Working alone, it would take Genny 1 more hour than it would take Ty. How long would it take each of them to complete the cleaning alone?

Answer 1

To solve this problem you have to work backwards. Here we know that it takes Genny and Ty 2 2/9 hours to do the work together. To find the time for these kinds of questions you have to find the reciprocal of the sum of each person's rate of work. For example, let's say Bob and Timmy can finish a painting job in 6 hours. That means that hourly, Bob and Timmy combined can do 1/6th of the work.

 

That means that if you add Bob's rate of work to Timmy's rate of work you would get 1/6 hourly. So a solution for this problem would be that Bob's rate of work is 1/12 per hour and Timmy's rate of work is 1/4 per hour. 

 

Now Back to the Original Problem. 

Genny's Time= t+1

 

Ty's Time= t

 

Together= 2 2/9 or 20/9 

 

 

Genny's Hourly Rate of Work is : 1/(t+1)

 

Ty's Hourly Rate of Work is: (1/t)

 

Combined Rate of Work Per Hour is: (1/ (20/9)= 9/20 

 

Now we can create an equation:

[1/(t+1)] + (1/t) = 9/20 

 

Putting the Denominator of this fraction into like terms would result in a Denominator of: (t+1)(t)(20) under each fraction. When you multiply everything out variables would cancel so your result would look like this:

 

20t/[20*(t+1)*t]+ 20t+20/[20*(t+1)*t]=9t²+9t/[20*(t+1)*t] 

 

Since the denominator is in like terms, you can just work with the numerator (like if you have 1/8+1/4=x/6 and you find the common denominator (24), then you can say 3/24 + 6/24= x/24 where you can just use the numerator alone to determine that 3+6=9.

 

So: 40t+20= 9t²+9t

Rewritten as: 9t²-31t-20=0. After you use the quadratic formula you will find that t=4. You disregard any negative answers of course since the time cannot be negative.

 

So your final answer would be that it takes Genny 5 hours alone and it takes Ty 4 hours alone. The answer makes sense because if you add the rate of work then Genny's Rate of work is 1/5 per hour. Ty's Rate of work is 1/4 per hour. If you convert each fraction using a common denominator you would get 4/20 for Genny and 5/20 for Ty. Add those together you get 9/20 as the total rate of work per hour. And then you get the reciprocal of that to determine that it takes 20/9 hours to complete the job. 

 

 

Answer 2

The key to story problems like this is that rates add.      Let G = Genny's rate  and T =  Ty's rate.

 

These rates can be thought of as B&Bs cleaned per hour.

 

From the story    G + T =   1/( 2 2/9) = 1/ (20/9) =    9/20

 

Also from the story   1/G =  1 +  1/T    {1/G is how long it would take Genny alone to clean the B&B}

 

Thus there are two equations and two unknowns.    The second equation can be simplified by multiplying both sides by the product   G T.   This results in:

 

   T =  T G + G

 

   The first equation can be solved for T in terms of G    i.e.  T = 9/20 -G

 

 This can be substituted into the revised form of the second equation to get:

   

   9/20 - G = (9/20 - G) G + G

 

This equation is quadratic in G.    Putting into standard form results in

 

   G² - (49/20) G + 9/20 = 0

 

 The quadratic formula can be used to solve it for G   .  The two solutions are

 

   G = 1/5  and  G =  9/4.     The second one leads to a negative value for  T, so must be rejected.

 

   with  G =  1/5    ,   T = 9/20 - 1/5 =   1/4.

 

   Thus it would take Genny  1/(1/5)) =  5 hours to do the job and  would take

  TY  1/(1/4) =  4 hours to do the job.